Set in Discrete Topology is Clopen

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Theorem

Let $T = \left({S, \tau}\right)$ be a discrete topological space.

$\forall U \subseteq S: U$ is both closed and open in $\left({S, \tau}\right)$.


Proof

Let $U \subseteq S$.

By definition of discrete topological space, $U \in \tau$.

By definition of closed set, $\complement_S \left({U}\right)$ is closed in $T$, where $\complement_S \left({U}\right)$ is the relative complement of $U$ in $S$.

But from Set Difference is Subset:

$\complement_S \left({U}\right) = S \setminus U \subseteq S$

and so:

$\complement_S \left({U}\right) \in \tau$

That is, $\complement_S \left({U}\right)$ is both closed and open in $T$.

Then by Relative Complement of Relative Complement:

$\complement_S \left({\complement_S \left({U}\right)}\right) = U$

which is seen to be both closed and open in $T$.

$\blacksquare$


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